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SCF CI calculations for vibrational eigenvalues and wavefunctions of systems exhibiting fermi resonance
1 October 1980
CHEMICAL PHYSICS LETTERS
Volume 75, number 1
SCF CI CALCULATIONS
FOR VIBRATIONAL
EIGENVALUES
AND WAVEFUNCTIONS
OF SYSTEMS EXHIBITING FERMI RESONANCE Todd C. THOMPSON and Donald G. TRUHLAR Departmertt of Chemwt~,
Unwersfty of Mnnesota,
Mmneaplrs,
Mhnesota
554.55, USA
Received 12 May 1980
It is shown that configuration-muung ulculat~ons unth configuratzons built from SCF wavefunctions can be used to provlde accurate and compact wavefunctions for systems m which the independent-mode approximation breaks down.
1. Introduction
for modal @ik IS
The self-conslstent field (SCF) method has long been a standard techmque for electronic structure problems m atoms and molecules, recently it has also been applied to problems of coupled vibratlonal modes [l-6]. A Ferrer resonance is a case of strong mixmg of two or more zero-order states of a coupled osclllator system that are accidentally degenerate or nearly degenerate 171. For such a case, the SCF method is expected to break doLtin. In this letter we demonstrate that It does break dawn for such a case and we show that good accuracy can be obtained by mlxmg just the resonant SCF states vanatlonally.
Hk@;k = ~;k@;k ,
2. Theory Consider a system vvlth two degrees of freedom x and y. Let the hamiltontan be H=h,(x,)
*‘l2@2)
+42(qr”2)
*
(1)
0)
where Hk is the Hartree operator Hk = h&k)
(4)
+ J,? (Xk)
and Jlk IS the self-consistent
field
JIyk~“k>=~dx,~~~(x,)~,~(~,,xk)Q;”’(xj)
-
0)
We solve the SCF equations by exp~dlng each modaI as a linear combmation of harmonic oscdlator (LCHO) elgenfunctions. We use P,? harmonic oscdlator functions m mode J; these functions are called the primitive basis set. Each SCF wavefunction JI,,,, may be considered as a basis function for the two-mode system. Then we consider the trial function
The SCF wavefunctlon for the v1v2 state is the modal product
Eachd+; is called an SCF configuration, and variational optimization of the cz!,F2 is called the SCF CI method. Thus yields the generalized eigenvaiue problem
+ “rV* = @‘;‘(~~)@(X21-
He, = E,Sc, ,
(2)
Notice that for electronic structure problems, we use an orbital product and interpret q as an independentparticle model For vibrational problems we call the single-mode wavefunctions modals and we interpret GJ as an independent-mode model. The SCF equation
(7)
where i = fv1u2),
1’ = @iv;)
Hig* = (3/ vlv*lHiJ/viV;) st,’
=
‘9v,v219viv;’
, 3
T
(8s) c9
87
CHEMKAL PHYSICS LETTERS
Volume 75, number 1
(10) and E/ =(~“lV21HI\Il”*y2)~
(11)
The number of configurations retained in (6) wil! be called N. The SCF approximation is N = 1. i.e., of, =Sii,.
3. System
As an example we consider a system with the following reduced hamiltonian [8], in reduced units: H =- f
a2ta.d t &W
+k ,22x~2
- d a2lay2 t
f o,:Y~
.V
+ k, 11.v3 ,
I I I 1 I
4. Calculations AlI matrix elements over primitive basis functions were evaluated using a program written by Harvey [9]. For most cases we used P,!‘j= 8; however, for v2 = 5, we used P$ = 10. Convergence checks showed that these values were sufficient to converge all energies in tables 1 and 2 to five significant figures. The eigenvalueeigenvector problem (3) was solved by the EISPACK subprogram RS and the eigenvalue-eigenvector problem (7) was solved by the EISPACK subprogram RSG.
(IZ)
The results for even states are in table 1; those for odd states are in table 2. For comparison the last column of each table gives the accurate eigenvalues determined by Cl calculations based on 60 configurations of each symmetry built up from primitive basis functions. The second last column of each table gives the eigenvalues obtained by the present calculations based
~___ .--
- --._ Vl”2 coo
VI”2 co4
“11’2 co2 _. . .
cvly2 12 _ ___ __
1.000
I .OOO I ,000 1.000
1.000
I
0.775 0.632
2
0.632 0.775
3
3 6
_ 3) ,ct’,l”fI c '1, q HH
I .OrJO 0.003 -0.064 I .norl 0.003 -0.002 -0.01 2 -0.004 --0.0I 2 n.on1.
1).004 0.775 -0.632 0.004 0.776 --0.631 0.005 -o.ons - 0.003
October 1980
5. Results and discussion
where wr = I .4, wj, = 0.7, k122 = -0.08. and kill = 0.0064. Because w.~ = 2w_,,and k122 is significant, this system exhibits Fermi resonance. Notice that the hamiltonian has a plane of symmetry: thus the vibrational wavefunctions may he divided into A’ states and A” states,
Table 1 A’ states ___.
1
0.719 0.481
0.651
0.502
0.730
iI)
0.017 0.003 0.002 0.650 0.207 0.731 ___
0.207
a) 0.632 0.715 3)
0.632 0.775
0.007 -0.007
0.003
0.720
3) 0.002
-0.481 -0.501
.
_-
VlV2 c20
E,, *u2 __--1.0493 2.4297 2.4497 3.7814 3.8316 1.000 3.8499 2.3899 2.4893 0.248 3.671 I 0.851 3.8329 -0.462 3.9583 1.0493 2.3899 2.4893 3) 1.0485 0.004 2.3899 -0.003 2.4893 0.248 3.6715 0.852 3.8330 0.46 I 3.9588 -__ __ ._._ -..---
._L
Accurate 1.0485 2.3859 2.4863 3.6584 3.8275 3.949s 2.3859 2.4863 3.6584 3.8275 3.949s 1.0485 2.3859 2.4863 1.0485 2.3859 2.4863 3.6584 3.8275 3.9495 ---
Volume 75, number 1
Table 2 A” states -._-I
._.. .“I”2
A’ _-__-..l
.
1
CNEMifAL.F’NYSICS LCTTERS
CO1
-._^--._--.-_. __I--L_.C. Y,“?
.._
.._. -
_._
CO5 - _
.
Cl3 .
1.000
1~000
1
1.a00
I
LOO0
1
1 2
1.000
0.715 -0.632
0.632 0.774 0.683 -0.582 -0.442
3 1.000 0.005 -0.004 t.000 0.004 -0.005 -0.023 -0.007 -0.021
3 6
_.I___
^.
0.006 0.175 -0.632 0.006 0.776 -0.631 0.003 -0.015 -0.003
0.663 0.235 0.710
n.311 0.779 0.545
0.030 ~.QO6 0.003 0.660 0.234 0.713
0.001 ~.0~7 0.006 0.311 0.780 -0.542
Fl1633 0.774 Z632 0.775 0.006 a) 0.004
. ..__
.
I .ono
1
-
.._.
“lk?
0.002 n.0~9 -0.009 0.686 -OS80 -0.44 1
1.7430 3.1092 3.1441 4.4458 4.5120 4.5452 3.0396 3.2135 4.2918 4SO8I 4.6702 1.7429 3.0397 3.2135 1.7404 3.0394 3.2133 4.2932 4.SOt36 4.7031
1.7404 3.0316 3.2070 4.2729 4.4960 4,6884 3.0316 3.2070 4.2729 4.4960 4.6884 1.7404 3.03t6 3.2070 1.7404 3.0316 3.2070 4.2729 4.4960 4.6884
.---._-I..._.
a) ,$fI’,Z I < 0.001. v , “2
on l-6 ~on~~uratians built up from SCF basis functions. The first column gives the number of SCF configurations, and the other columns give the coefficients of the SCF ~on~gurations in the SCF CI wavefullction. The identity of the state is clear from the largest coeforients. An accuracy of 0.010 in reduced units will be con. sidered roughly comparable to +lO~m-~ and therefore acceptable. For the (0,O) and (0, I) states, which do not exhibit Fermi resonance. we obtain accuracies of 0.001 and 0,003. Consider next the lowest Fermi resonance, involving the (0, ?) and (1.0) states. The SCF states have errors of 0.044 and 0.036. Mixing these two states decreases the errors to 0.004 and 0.003, so that the errors are now ~omparabJc to the errors in the non-Fermi-resonant states. Furthermore, the mixing increases the splitting of these two states from 0.020 to 0.099, in good agreement with tire accurate value of 0.100. Mixing in the ground-state SCF co~~guration or the next three SCF con~gurations of this symmetry has only a small effect. These results seem to indicate that modal reoptimization is not a sJgnJ~~ant effect, i.e., the best medals for the two-configuration wave-
sanction must be very similar to those for the onecon~guration calculation, even though there is strong mixing. We obtain con~parable results for the next Fermi resonance, the one based on the (0,3) and {1, I) states. fn this case the SCF sp~jttjn~of 0.035 is increased to 0.174, in good agreement with the accurate value of 0.175. The next two Fermi res~)l~an~esinvulvc 3 states each. Again we find a large increase in ihe splitting in mixing the strongly interacting SCF states, and again we find that the resulting wavefunctions are reasonably stable to mixing in lower-energy SCF states that are not part of the Fermi resonance. For the A” triad the errurs in the SCF ejgenvalues are 0.123,0.023, and 0.100, and these errors are decreased to 0,013,0.006, and 0.008 by the mixing of tire three dominant configurations. Mixing the six lowest-energy SCF functions of even symmetry changes the results only in the fifth signi~~ant figure. The errors arc quajltit:~tively similar to the A” triad. it is jnteresting to notice the great sir~JJarJty Jn tJre pattern of coefficients for the A” case to that for the A’ ease. 80
Volume 75, number 1
CHEMICAL PHYSICS LETTERS
1 October 1980
6. Conclusion
References
We have shown that even m the case of strongly interacting states, when the mdependent-mode model breaks down, the SCF method 1s stfl very useful for generating configurations. The use of SCF configurations in a configuration-mixing calculation mvolvmg only the strongly interacting configurations provide a reasonably accurate, yet compact representation of the Fermi resonance states.
[I] G.C. Camey, L.L. Sprandel and C.W. Kern, Advan. Chem. Phys. 37 (1978) 305. 121 __, I. Bowman, J. Chem. Phys. 68 (1978) 608. 131 J.M. Bowman, K. Christoffel and F.M. Tobm, 1. Phys. Chem. 83 (1979) 905. r41 M. Cohen, S. Greita and R P. McEachran. Chem. Phys. Letters 60 (1979) 445. [51 R.B. Gerber and M.A. Ratner, Chem. Phys Letters 68 (1979) 195. [61 F.L. Tobm and J.M. Bowman, Chem. Phys. 47 (1980) 151. 171 S. Cal&no, Vibrational states (Wiley, New York, 1976). and RA. Marcus, J. Chem. 181 D.W. Noid, M.L. Koaykowski Phys. 71 (1979) 2864. [91 N.M. Harvey, Ph.D. Thesis, Umverslty of Minnesota, Mlnneapohs (1979).
Acknowledgement This work was supported m part by the National Science Foundation under grant no. CHE77-27415.
90
CHEMICAL PHYSICS LETTERS
Volume 75, number 1
SCF CI CALCULATIONS
FOR VIBRATIONAL
EIGENVALUES
AND WAVEFUNCTIONS
OF SYSTEMS EXHIBITING FERMI RESONANCE Todd C. THOMPSON and Donald G. TRUHLAR Departmertt of Chemwt~,
Unwersfty of Mnnesota,
Mmneaplrs,
Mhnesota
554.55, USA
Received 12 May 1980
It is shown that configuration-muung ulculat~ons unth configuratzons built from SCF wavefunctions can be used to provlde accurate and compact wavefunctions for systems m which the independent-mode approximation breaks down.
1. Introduction
for modal @ik IS
The self-conslstent field (SCF) method has long been a standard techmque for electronic structure problems m atoms and molecules, recently it has also been applied to problems of coupled vibratlonal modes [l-6]. A Ferrer resonance is a case of strong mixmg of two or more zero-order states of a coupled osclllator system that are accidentally degenerate or nearly degenerate 171. For such a case, the SCF method is expected to break doLtin. In this letter we demonstrate that It does break dawn for such a case and we show that good accuracy can be obtained by mlxmg just the resonant SCF states vanatlonally.
Hk@;k = ~;k@;k ,
2. Theory Consider a system vvlth two degrees of freedom x and y. Let the hamiltontan be H=h,(x,)
*‘l2@2)
+42(qr”2)
*
(1)
0)
where Hk is the Hartree operator Hk = h&k)
(4)
+ J,? (Xk)
and Jlk IS the self-consistent
field
JIyk~“k>=~dx,~~~(x,)~,~(~,,xk)Q;”’(xj)
-
0)
We solve the SCF equations by exp~dlng each modaI as a linear combmation of harmonic oscdlator (LCHO) elgenfunctions. We use P,? harmonic oscdlator functions m mode J; these functions are called the primitive basis set. Each SCF wavefunction JI,,,, may be considered as a basis function for the two-mode system. Then we consider the trial function
The SCF wavefunctlon for the v1v2 state is the modal product
Eachd+; is called an SCF configuration, and variational optimization of the cz!,F2 is called the SCF CI method. Thus yields the generalized eigenvaiue problem
+ “rV* = @‘;‘(~~)@(X21-
He, = E,Sc, ,
(2)
Notice that for electronic structure problems, we use an orbital product and interpret q as an independentparticle model For vibrational problems we call the single-mode wavefunctions modals and we interpret GJ as an independent-mode model. The SCF equation
(7)
where i = fv1u2),
1’ = @iv;)
Hig* = (3/ vlv*lHiJ/viV;) st,’
=
‘9v,v219viv;’
, 3
T
(8s) c9
87
CHEMKAL PHYSICS LETTERS
Volume 75, number 1
(10) and E/ =(~“lV21HI\Il”*y2)~
(11)
The number of configurations retained in (6) wil! be called N. The SCF approximation is N = 1. i.e., of, =Sii,.
3. System
As an example we consider a system with the following reduced hamiltonian [8], in reduced units: H =- f
a2ta.d t &W
+k ,22x~2
- d a2lay2 t
f o,:Y~
.V
+ k, 11.v3 ,
I I I 1 I
4. Calculations AlI matrix elements over primitive basis functions were evaluated using a program written by Harvey [9]. For most cases we used P,!‘j= 8; however, for v2 = 5, we used P$ = 10. Convergence checks showed that these values were sufficient to converge all energies in tables 1 and 2 to five significant figures. The eigenvalueeigenvector problem (3) was solved by the EISPACK subprogram RS and the eigenvalue-eigenvector problem (7) was solved by the EISPACK subprogram RSG.
(IZ)
The results for even states are in table 1; those for odd states are in table 2. For comparison the last column of each table gives the accurate eigenvalues determined by Cl calculations based on 60 configurations of each symmetry built up from primitive basis functions. The second last column of each table gives the eigenvalues obtained by the present calculations based
~___ .--
- --._ Vl”2 coo
VI”2 co4
“11’2 co2 _. . .
cvly2 12 _ ___ __
1.000
I .OOO I ,000 1.000
1.000
I
0.775 0.632
2
0.632 0.775
3
3 6
_ 3) ,ct’,l”fI c '1, q HH
I .OrJO 0.003 -0.064 I .norl 0.003 -0.002 -0.01 2 -0.004 --0.0I 2 n.on1.
1).004 0.775 -0.632 0.004 0.776 --0.631 0.005 -o.ons - 0.003
October 1980
5. Results and discussion
where wr = I .4, wj, = 0.7, k122 = -0.08. and kill = 0.0064. Because w.~ = 2w_,,and k122 is significant, this system exhibits Fermi resonance. Notice that the hamiltonian has a plane of symmetry: thus the vibrational wavefunctions may he divided into A’ states and A” states,
Table 1 A’ states ___.
1
0.719 0.481
0.651
0.502
0.730
iI)
0.017 0.003 0.002 0.650 0.207 0.731 ___
0.207
a) 0.632 0.715 3)
0.632 0.775
0.007 -0.007
0.003
0.720
3) 0.002
-0.481 -0.501
.
_-
VlV2 c20
E,, *u2 __--1.0493 2.4297 2.4497 3.7814 3.8316 1.000 3.8499 2.3899 2.4893 0.248 3.671 I 0.851 3.8329 -0.462 3.9583 1.0493 2.3899 2.4893 3) 1.0485 0.004 2.3899 -0.003 2.4893 0.248 3.6715 0.852 3.8330 0.46 I 3.9588 -__ __ ._._ -..---
._L
Accurate 1.0485 2.3859 2.4863 3.6584 3.8275 3.949s 2.3859 2.4863 3.6584 3.8275 3.949s 1.0485 2.3859 2.4863 1.0485 2.3859 2.4863 3.6584 3.8275 3.9495 ---
Volume 75, number 1
Table 2 A” states -._-I
._.. .“I”2
A’ _-__-..l
.
1
CNEMifAL.F’NYSICS LCTTERS
CO1
-._^--._--.-_. __I--L_.C. Y,“?
.._
.._. -
_._
CO5 - _
.
Cl3 .
1.000
1~000
1
1.a00
I
LOO0
1
1 2
1.000
0.715 -0.632
0.632 0.774 0.683 -0.582 -0.442
3 1.000 0.005 -0.004 t.000 0.004 -0.005 -0.023 -0.007 -0.021
3 6
_.I___
^.
0.006 0.175 -0.632 0.006 0.776 -0.631 0.003 -0.015 -0.003
0.663 0.235 0.710
n.311 0.779 0.545
0.030 ~.QO6 0.003 0.660 0.234 0.713
0.001 ~.0~7 0.006 0.311 0.780 -0.542
Fl1633 0.774 Z632 0.775 0.006 a) 0.004
. ..__
.
I .ono
1
-
.._.
“lk?
0.002 n.0~9 -0.009 0.686 -OS80 -0.44 1
1.7430 3.1092 3.1441 4.4458 4.5120 4.5452 3.0396 3.2135 4.2918 4SO8I 4.6702 1.7429 3.0397 3.2135 1.7404 3.0394 3.2133 4.2932 4.SOt36 4.7031
1.7404 3.0316 3.2070 4.2729 4.4960 4,6884 3.0316 3.2070 4.2729 4.4960 4.6884 1.7404 3.03t6 3.2070 1.7404 3.0316 3.2070 4.2729 4.4960 4.6884
.---._-I..._.
a) ,$fI’,Z I < 0.001. v , “2
on l-6 ~on~~uratians built up from SCF basis functions. The first column gives the number of SCF configurations, and the other columns give the coefficients of the SCF ~on~gurations in the SCF CI wavefullction. The identity of the state is clear from the largest coeforients. An accuracy of 0.010 in reduced units will be con. sidered roughly comparable to +lO~m-~ and therefore acceptable. For the (0,O) and (0, I) states, which do not exhibit Fermi resonance. we obtain accuracies of 0.001 and 0,003. Consider next the lowest Fermi resonance, involving the (0, ?) and (1.0) states. The SCF states have errors of 0.044 and 0.036. Mixing these two states decreases the errors to 0.004 and 0.003, so that the errors are now ~omparabJc to the errors in the non-Fermi-resonant states. Furthermore, the mixing increases the splitting of these two states from 0.020 to 0.099, in good agreement with tire accurate value of 0.100. Mixing in the ground-state SCF co~~guration or the next three SCF con~gurations of this symmetry has only a small effect. These results seem to indicate that modal reoptimization is not a sJgnJ~~ant effect, i.e., the best medals for the two-configuration wave-
sanction must be very similar to those for the onecon~guration calculation, even though there is strong mixing. We obtain con~parable results for the next Fermi resonance, the one based on the (0,3) and {1, I) states. fn this case the SCF sp~jttjn~of 0.035 is increased to 0.174, in good agreement with the accurate value of 0.175. The next two Fermi res~)l~an~esinvulvc 3 states each. Again we find a large increase in ihe splitting in mixing the strongly interacting SCF states, and again we find that the resulting wavefunctions are reasonably stable to mixing in lower-energy SCF states that are not part of the Fermi resonance. For the A” triad the errurs in the SCF ejgenvalues are 0.123,0.023, and 0.100, and these errors are decreased to 0,013,0.006, and 0.008 by the mixing of tire three dominant configurations. Mixing the six lowest-energy SCF functions of even symmetry changes the results only in the fifth signi~~ant figure. The errors arc quajltit:~tively similar to the A” triad. it is jnteresting to notice the great sir~JJarJty Jn tJre pattern of coefficients for the A” case to that for the A’ ease. 80
Volume 75, number 1
CHEMICAL PHYSICS LETTERS
1 October 1980
6. Conclusion
References
We have shown that even m the case of strongly interacting states, when the mdependent-mode model breaks down, the SCF method 1s stfl very useful for generating configurations. The use of SCF configurations in a configuration-mixing calculation mvolvmg only the strongly interacting configurations provide a reasonably accurate, yet compact representation of the Fermi resonance states.
[I] G.C. Camey, L.L. Sprandel and C.W. Kern, Advan. Chem. Phys. 37 (1978) 305. 121 __, I. Bowman, J. Chem. Phys. 68 (1978) 608. 131 J.M. Bowman, K. Christoffel and F.M. Tobm, 1. Phys. Chem. 83 (1979) 905. r41 M. Cohen, S. Greita and R P. McEachran. Chem. Phys. Letters 60 (1979) 445. [51 R.B. Gerber and M.A. Ratner, Chem. Phys Letters 68 (1979) 195. [61 F.L. Tobm and J.M. Bowman, Chem. Phys. 47 (1980) 151. 171 S. Cal&no, Vibrational states (Wiley, New York, 1976). and RA. Marcus, J. Chem. 181 D.W. Noid, M.L. Koaykowski Phys. 71 (1979) 2864. [91 N.M. Harvey, Ph.D. Thesis, Umverslty of Minnesota, Mlnneapohs (1979).
Acknowledgement This work was supported m part by the National Science Foundation under grant no. CHE77-27415.
90